A minimal HRT counterexample at the sharp Gelfand--Shilov threshold

Abstract

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil--Ramanathan--Topiwala conjecture by constructing twelve linearly dependent time--frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand--Shilov class 𝒮₁¹(ℝ); their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in 𝒮₁¹(ℝ). Moreover, if 0≠ f∈𝒮_s^σ(ℝ) and s<1 or σ<1, then every finite system of distinct time--frequency shifts of f is linearly independent. Thus four is the least possible cardinality and (1,1) is the sharp coordinatewise threshold within the Roumieu Gelfand--Shilov scale.

Regularity under Integral Closure: A General Criterion and a Sharp Counterexample
Soumyadeep Misra
arXiv9 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements
Samuel J. Ferguson, Joshua Iosevich
arXiv30 Oct 2026PreprintCC BY-NC-SA 4.0Full textAnalysis & PDEs Combinatorics & discrete mathematicsarXiv
Minimal transitive factorizations supported on quasi-threshold graphs
Cordelia Yuqiao Li, Ricky Ini Liu
arXiv25 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
The Sharp Ellipticity Threshold for W^1,1-Regularity
Franz Gmeineder
arXiv24 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
The sharp and non-sharp threshold phenomenon in buffered and Go-or-Grow systems
Ruben Taieb, Arnaud Ducrot
HAL20 Sep 2026PreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsHAL

Cite

BibTeX RIS

BibTeX

@misc{oussa2026minimal,
  title = {{A minimal HRT counterexample at the sharp Gelfand--Shilov threshold}},
  author = {Vignon Oussa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27774},
  eprint = {2609.27774},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27774},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A minimal HRT counterexample at the sharp Gelfand--Shilov threshold
AU  - Vignon Oussa
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27774
UR  - https://arxiv.org/abs/2609.27774
AB  - In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil--Ramanathan--Topiwala conjecture by constructing twelve linearly dependent time--frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand--Shilov class 𝒮_1^1(ℝ); their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in 𝒮_1^1(ℝ). Moreover, if 0≠ f∈𝒮_s^σ(ℝ) and s<1 or σ<1, then every finite system of distinct time--frequency shifts of f is linearly independent. Thus four is the least possible cardinality and (1,1) is the sharp coordinatewise threshold within the Roumieu Gelfand--Shilov scale.
ER  -

Abstract

Related

Cite